Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Random subgroups of branch groups

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In super strongly fractal branch profinite groups, k Haar-random elements almost surely generate a free group of rank k whose action on the tree boundary is free.

desk verdict The main theorem is likely right, but the proof of Theorem 1 misapplies the independence lemma to random vertices—a genuine gap that needs fixing. read the letter →

arxiv 2508.20082 v1 pith:C6DKWFWU submitted 2025-08-27 math.GR

Jorge Fariña-Asategui , Santiago Radi This is my paper
classification math.GR MSC 20E0820E1828C10
keywords randomsubgroupsbranchgroupsprofiniteHaarmeasurefreeboundaryactionfixed-pointproportionself-similar
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that in any super strongly fractal branch profinite group acting on a regular rooted tree, k independent Haar-random elements generate, almost surely, a free subgroup of rank k that acts on the boundary of the tree with no fixed points. This sharpens an earlier theorem of Abért that only guaranteed an almost free action, and it generalizes a result of Abért and Virág that was specific to iterated wreath products. The key step is a new probabilistic independence lemma: sections of a random element at sufficiently separated vertices behave like independent Haar-random elements, because section maps preserve Haar measure. If correct, the theorem applies to the closures of the first Grigorchuk group, periodic GGS-groups, the iterated monodromy group of z^2+i, the Hanoi towers group on three pegs, and other well-studied branch groups. It also implies that, for k at least 2, the boundary action of such a random subgroup is almost surely not amenable, and it yields a negative answer to a question about fixed-point proportions of random subgroups.

What carries the argument

The proof is carried by Lemma 3.1: in a super strongly fractal group of finite type of depth D, the sections of a single Haar-random element at any set of vertices that are pairwise not (D−1)-cousins form a collection of independent Haar-random elements. This lemma replaces the Galton–Watson probabilistic structure that made the iterated wreath product case tractable; for general super strongly fractal branch groups, the group is, by Theorem 2.4, of finite type, and the branching is strong enough that sections at far-apart vertices decorrelate. The independence lemma is powered by Theorem 2.5, which says that the section maps φ_v from G to G preserve Haar measure, and by the explicit branchi

What would settle it

For the closure of the first Grigorchuk group, compute the probability that a fixed reduced word in two random elements fixes a leaf at level n; if the limiting probability is positive for any non-trivial word, the boundary action is not free and Theorem 1 is false. The same check can be run for any candidate super strongly fractal branch group.

Watch

Extended reading notes

Core claim

Theorem 1: Let G be a super strongly fractal branch profinite group acting on a d-regular rooted tree T. Then almost surely, for every k at least 1, a k-generated random subgroup of G is a free group of rank k acting freely on the boundary of T. This upgrades Abért's earlier 'almost free' conclusion to genuine freeness of the action and extends the iterated-wreath-product result of Abért and Virág to the whole class of super strongly fractal branch groups. A companion result, Theorem 3, shows that if G is topologically finitely generated with fixed-point proportion zero, then a k-generated random subgroup has null fixed-point proportion with probability tending to 1 as k tends to infinity.

Load-bearing premise

The key premise is that, in these groups, looking at the part of an automorphism below a fixed vertex does not distort the random distribution of the whole element; the whole argument collapses if that measure preservation fails.

Editorial extensions

If this is right

  • For every example listed in the introduction, including the closures of the first Grigorchuk group, periodic GGS-groups, the iterated monodromy group of z^2+i, and the Hanoi towers group on three pegs, k-random subgroups are free groups of rank k whose boundary actions are free.
  • Corollary 2: for k at least 2, the action of a k-random subgroup on the tree boundary is almost surely not amenable, because the subgroup is a non-abelian free group and a group admitting a free amenable action must be amenable.
  • Theorem 3: if G is topologically finitely generated and FPP(G) = 0, then the probability that a k-random subgroup has FPP = 0 tends to 1 as k tends to infinity; Corollary 4 gives the explicit lower bound ∏_{j=0}^{d−1} (1 − 1/p^{k−j}) for pro-p subgroups of the p-adic automorphism group W_p.
  • Section 4 shows that neither hypothesis in Theorem 1 can be dropped: an elementary abelian super strongly fractal group has finite (hence non-free-acting) random subgroups, and a regular branch group of finite type that is not super strongly fractal can have positive fixed-point proportion.
  • The counterexample in Section 4.1 answers a question of Radi negatively: a group with null fixed-point proportion can have a finitely generated random subgroup with positive fixed-point proportion with probability 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The independence lemma is likely to extend beyond branch groups: any self-similar closed group whose section maps are measure-preserving and whose branching at separated vertices is sufficiently rigid should satisfy the same independence property; this could be tested on groups with weaker branching conditions.
  • The proof should be amenable to quantitative refinement for explicit groups: tracking the level N introduced in the induction gives an explicit bound on the tree depth one must inspect before a given word w moves a random tuple, which could be computed for the Grigorchuk group or the Hanoi towers group.
  • In arithmetic dynamics, if the arboreal Galois group attached to a rational function is super strongly fractal and branch, Theorem 1 implies that random tuples of Galois automorphisms act freely on the preimage tree, which may feed into density statements for periodic points over finite fields.
  • The reformulated Question 1.2 suggests a conjecture that non-locally finite closed subgroups with FPP = 0 have k-random subgroups with FPP = 0 for each fixed k; Theorem 3 is only a partial answer in the regime where k grows, so proving the fixed-k case would require new tools beyond the present argument.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies random subgroups of super strongly fractal branch profinite groups acting on a regular rooted tree. The main result, Theorem 1, asserts that k independent Haar-random elements almost surely generate a free group of rank k that acts freely on the boundary of the tree. The proof strategy is to prove a probabilistic independence lemma for sections of a random element at pairwise non-cousin vertices (Lemma 3.1), then to argue by induction on word length that a random evaluation of a word fixes no end. The paper also proves Theorem 3, about the fixed-point proportion of random subgroups of topologically finitely generated groups with FPP(G)=0, and gives examples showing that the hypotheses of Theorem 1 cannot simply be dropped.

Significance. If the main theorem is correct, it is a substantial generalization of the Abért–Virág result for iterated wreath products to a broad class of branch groups, with attractive consequences for amenability of boundary actions and for arithmetic-dynamics questions. The independence lemma is a potentially useful new tool, and the fixed-point-proportion results are of independent interest. However, the proof of Theorem 1 as written contains a serious gap: Lemma 3.1 is applied to a collection of vertices that is itself random, which the lemma does not justify. The main theorem is therefore not established in the submitted form.

major comments (2)
  1. [Section 3.2, proof of Theorem 1] Lemma 3.1 is stated and proved only for a fixed set V ⊆ L_n. In the induction step of Theorem 1, the vertices v_i = v · w_i(g_1, ..., g_k) are defined after the random tuple is sampled; the set {v_1, ..., v_{ℓ-1}} is a function of (g_1, ..., g_k). Lemma 3.1 cannot be applied to such a random set. This is not a purely formal objection: take G = Aut(T_2), which is super strongly fractal and branch, and let w = x_1^2, v ∈ L_1. Then w(g)|_v = g|_v · g|_{v·g}. Projecting to the depth-2 quotient C2 ≀ C2, one computes P(w(g)|_v = 1) = 3/4, while a Haar-random section of Aut(T_2) satisfies P = 1/2. Hence the assertion that w(g)|_v is a Haar-random element is false, and the induction step relying on Theorem 2.7 breaks down. The theorem may still be true, but this proof does not establish it.
  2. [Section 3.1, Lemma 3.1, Equation (3.3)] Equation (3.3) as printed places the kernel factor in the denominator: it reads |π_n(G)| · ∏ #A_v / (|π_{n+m}(G)| · |ker φ|). The preceding counting argument gives the kernel factor in the numerator: the number of lifts of each configuration is |ker φ|, so the correct expression is |π_n(G)| · (∏ #A_v) · |ker φ| / |π_{n+m}(G)|. The subsequent comparison with Equation (3.1) uses the corrected form. This appears to be a typographical slip, but it must be fixed for the proof of Lemma 3.1 to be formally correct.
minor comments (4)
  1. [Section 4.2] The sentence 'This implies that a random subgroup does not act freely on the boundary ∂T with probability 1' is overstated. If FPP(G_P^Q) = p > 0, then the probability that a k-generated random subgroup fails to act freely is at most 1 - (1-p)^k, which is strictly less than 1 unless p = 1. The text should say 'with positive probability' unless the stronger statement is proved.
  2. [Section 3.3, Theorem 3.2] Theorem 3.2 is attributed to [16, Theorem 1] in the parenthetical, but the surrounding text says the result is proved in [6]. The citation should be corrected to the Borovik–Pyber–Shalev paper [6].
  3. [Lemma 3.1 / Section 1] For D = 1, the condition 'not (D-1)-cousins' is undefined, since Definition 2.1 requires m ≥ 1. Since Theorem 1 is claimed for iterated wreath products (D = 1), the paper should either define the m = 0 case, or treat D = 1 separately using the known Abért–Virág result.
  4. [Throughout] Minor wording/typos: 'measuring-preserving' should be 'measure-preserving'; in the proof of Theorem 3, 'FPP(Hk) = FPP(Hk)' should presumably be 'FPP(H_k) = FPP(\overline{H_k})'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central derivation is independent; self-citations are non-equivalent prior results.

full rationale

The derivation chain of Theorem 1 is not circular. The freeness of the random subgroup is imported from Abért's theorem (Corollary 2.6), and the boundary freeness is reduced to (i) FPP(G)=0 for a single Haar element (Theorem 2.7, cited from [16]) and (ii) independence of sections at sufficiently separated vertices (Lemma 3.1). Lemma 3.1 is proved in the paper by a counting argument using the structure of finite-type branch groups; it does not assume the conclusion of Theorem 1. The cited results [13], [15], [16] are prior results by the same authors, but each is a parameter-free statement about fractal/branch/super-strongly-fractal groups (section-map measure preservation, equivalence of branch-type notions, fixed-point proportion zero) and none has the target theorem—that k-generated random subgroups are free and act freely—as an assumption or conclusion. There are no fitted parameters and no quantity is defined in terms of the quantity being predicted. The proof's application of Lemma 3.1 to vertices v_i that depend on the random tuple is a potential gap (Lemma 3.1 is stated for a fixed vertex set), and the repeated-letter case (e.g., w=x_1^2) may break the claimed independence; but this is a correctness risk, not a circular reduction of the conclusion to the hypotheses. Therefore no circularity step is identified; score 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted parameters; the paper introduces no new entities. The theorem rests on known structure theory of branch groups and on four prior results, three by the present authors, used as black boxes.

assumptions (7)
  • standard math Unique normalized Haar measure on profinite groups and cone-set formula mu(C_A)=#A/|pi_n(G)|.
    Probability framework of the paper; used throughout Section 3.
  • domain assumption Section maps phi_v preserve Haar measure on fractal groups (Theorem 2.5, [13]).
    Used in Equation (3.1) of Lemma 3.1 to equate mu(phi_v^{-1}(C_A)) with mu(C_A); if false, the independence lemma collapses.
  • domain assumption For fractal closed subgroups of Aut(T), finite type, regular branch, and branch are equivalent (Theorem 2.4, [15]).
    Used at the start of Section 3 to assume a super strongly fractal branch profinite group has finite type of depth D and is regular branch over St_G(D-1).
  • domain assumption Super strongly fractal groups have zero fixed-point proportion (Theorem 2.7, [16]).
    Base case of the induction in Theorem 1; also used in Section 4 to contrast with positive FPP examples.
  • domain assumption Abért's theorem: k independent Haar-random elements in a compact separating-action group generate a free subgroup, almost freely in the transitive case (Corollary 2.6).
    Gives freeness of the abstract subgroup; Theorem 1 adds freeness of the boundary action.
  • domain assumption Borovik-Pyber-Shalev random generation theorem (Theorem 3.2), with the stated correction to 2-zeta(s).
    Used in Theorem 3 to bound the probability that random elements project onto all finite quotients.
  • domain assumption Regular branch product decomposition psi(St(n)) = St(D-1)^{d^{n-D+1}} (Theorem 2.3, [37], [21]).
    Used inside Lemma 3.1 to construct group elements realizing arbitrary prescribed sections.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Random subgroups of branch groups." pith.science (2026). https://pith.science/paper/C6DKWFWU

@misc{pith2026250820082,
  author       = {Pith},
  title        = {Pith review of: Random subgroups of branch groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6DKWFWU}},
  note         = {Machine review of arXiv:2508.20082}
}
read the original abstract

We show that independent Haar-random elements in a super strongly fractal branch profinite group generate a free subgroup acting freely on the boundary of the tree. This improves a previous result of Ab\'ert (2005) for weakly branch profinite groups, where independent random elements were shown to generate free subgroups acting only almost freely on the boundary. Our result also generalizes the analogous result of Ab\'ert and Vir\'ag (2005) for iterated wreath products.

Figures

Figures reproduced from arXiv: 2508.20082 by the authors.

Figure 1
Figure 1. An illustration of vertices which are not m-cousins: w1 and w2 are at least m levels below v1 and v2 respectively. 2.2. Groups acting on regular rooted trees. The group of automorphisms of T, denoted Aut(T), is the group of graph automorphisms of T, i.e. functions fixing the root and preserving adjacency. In particular Aut(T) acts by permuting vertices at the same level of T. The following observation will be useful… view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Groups of finite type: classification and structural properties

    math.GR 2025-09 conditional novelty 7.0 of 10

    For many groups of finite type, topological finite generation, just-infiniteness and strong completeness are equivalent, and new algorithms classify the groups up to isomorphism on binary and ternary trees.

Reference graph

Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [16]

    Fariña-Asategui, S

    J. Fariña-Asategui, S. Radi, On the fixed-point proportion of self-similar groups, arXiv preprint: 2503.00185

    arXiv

  2. [6]

    Borovik, L

    A.V. Borovik, L. Pyber and A. Shalev, Maximal subgroups in finite and profinite groups, Trans. Amer. Math. Soc.348 (9) (1996), 3745–3761

    work page 1996

  3. [1]

    Abért, Group laws and free subgroups in topological groups,Bull

    M. Abért, Group laws and free subgroups in topological groups,Bull. London Math. Soc.37 (2005), 525–534

    work page 2005

  4. [2]

    Abért and B

    M. Abért and B. Virág, Dimension and randomness in groups acting on rooted trees,J. Amer. Math. Soc. 18 (2005), 157–192

    work page 2005

  5. [3]

    Bartholdi and V

    L. Bartholdi and V. Nekrashevych, Iterated monodromy groups of quadratic polynomials, I, Groups Geom. Dyn.2 (2008), 309–336

    work page 2008

  6. [4]

    Benedetto, P

    R. Benedetto, P. Ingram, R. Jones, M. Manes, J.H. Silverman, and T.J. Tucker, Current trends and open problems in arithmetic dynamics,Bull. Amer. Math. Soc.56 (2019), 611– 685

    work page 2019

  7. [5]

    Bridy, R

    A. Bridy, R. Jones, G. Kelsey, and R. Lodge, Iterated monodromy groups of rational functions and periodic points over finite fields,Math. Ann. 390(1) (2024), 439–475

    work page 2024

  8. [7]

    Bowen, Non-abelian free group actions: Markov processes, the Abramov- Rohlin formula and Yuzvinskii’s formula,Ergodic Theory Dynam

    L. Bowen, Non-abelian free group actions: Markov processes, the Abramov- Rohlin formula and Yuzvinskii’s formula,Ergodic Theory Dynam. Systems30 (6) (2010), 1629–1663

    work page 2010

Show all 38 references
  1. [8]

    Burness, M.W

    T.C. Burness, M.W. Liebeck and A. Shalev, Generation and random generation: From simple groups to maximal subgroups,Adv. Math. 248 (2013), 59–95

    2013

  2. [9]

    Dixon, The probability of generating the symmetric group,Math

    J.D. Dixon, The probability of generating the symmetric group,Math. Z.110(1969), 199–205

    1969

  3. [10]

    van Douwen, Measures invariant under actions of F2,Topology Appl.34 (1990), 53–68

    E.K. van Douwen, Measures invariant under actions of F2,Topology Appl.34 (1990), 53–68

    1990

  4. [11]

    Evans, Eigenvalues of random wreath products,Electron

    S.N. Evans, Eigenvalues of random wreath products,Electron. J. Probab.7 (2002), 1–15

    2002

  5. [12]

    Erdős and P

    P. Erdős and P. Turan, On some problems of a statistical group-theory,I. Wahrscheinlichkeit- stheorie und Verw. Gebiete4 (1965), 175–186

    1965

  6. [13]

    Fariña-Asategui, Cyclicity, hypercyclicity and randomness in self-similar groups,Monatsh

    J. Fariña-Asategui, Cyclicity, hypercyclicity and randomness in self-similar groups,Monatsh. Math. 207 (2025), 275–292

    2025

  7. [14]

    Fariña-Asategui, Markov processes associated to fractal branch groups, arXiv preprint: 2505.21134

    J. Fariña-Asategui, Markov processes associated to fractal branch groups, arXiv preprint: 2505.21134

    arXiv

  8. [15]

    Fariña-Asategui, Restricted Hausdorff spectra ofq-adic automorphisms, Adv

    J. Fariña-Asategui, Restricted Hausdorff spectra ofq-adic automorphisms, Adv. Math. 472 (2025), 110294

    2025

  9. [17]

    Fernández-Alcober and A

    G.A. Fernández-Alcober and A. Zugadi-Reizabal, GGS-groups: order of congruence quotients and Hausdorff dimension,Trans. Amer. Math. Soc.366 (4) (2014), 1993–2017

    2014

  10. [18]

    Fried, Field Arithmetic,Springer Berlin Heidelberg, 11 (2023), xxxi–827

    M.D. Fried, Field Arithmetic,Springer Berlin Heidelberg, 11 (2023), xxxi–827

    2023

  11. [19]

    Glasner and N

    Y. Glasner and N. Monod, Amenable actions, free products and a fixed point property,Bull. London Math. Soc.39 (2007), 138–150

    2007

  12. [20]

    Grigorchuk, On Burnside’s problem for periodic groups,Funktsional

    R.I. Grigorchuk, On Burnside’s problem for periodic groups,Funktsional. Anal. i Prilozhen 14 (1) (1980), 53–54

    1980

  13. [21]

    Grigorchuk, Solved and unsolved problems around one group, in:Infinite groups: Geo- metric, Combinatorial and Dynamical Aspects, Progr

    R.I. Grigorchuk, Solved and unsolved problems around one group, in:Infinite groups: Geo- metric, Combinatorial and Dynamical Aspects, Progr. Math.248, 2005

    2005

  14. [22]

    Grigorchuk and V

    R.I. Grigorchuk and V. Nekrashevych, Amenable actions of non-amenable groups,J. Math. Sci. 140 (2007), 391–397

    2007

  15. [23]

    Grigorchuk, V

    R.I. Grigorchuk, V. Nekrashevych and Z. Šunić, Hanoi Towers groups,Oberwolfach Rep.19 (2006), 11–14

    2006

  16. [24]

    Jaikin-Zapirain and L

    A. Jaikin-Zapirain and L. Pyber, Random generation of finite and profinite groups and group enumeration, Ann. Math. 173 (2011), 769–814. RANDOM SUBGROUPS OF BRANCH GROUPS 15

    2011

  17. [25]

    Jones, Fixed-point-free elements of iterated monodromy groups,Trans

    R. Jones, Fixed-point-free elements of iterated monodromy groups,Trans. Amer. Math. Soc. 367(3) (2015), 2023–2049

    2015

  18. [26]

    Jones, Galois representations from pre-image trees: an arboreal survey,Publ

    R. Jones, Galois representations from pre-image trees: an arboreal survey,Publ. Math. Be- sançon, Algèbre Théor. Nr.(2013), 107–136

    2013

  19. [27]

    Jones, Iterated Galois towers, their associated martingales, and thep-adic Mandelbrot set, Compos

    R. Jones, Iterated Galois towers, their associated martingales, and thep-adic Mandelbrot set, Compos. Math. 143(5) (2007), 1108–1126

    2007

  20. [28]

    Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J

    R. Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. London Math. Soc.78(2) (2008), 523–544

    2008

  21. [29]

    J. Juul, P. Kurlberg, K. Madhu, and T.J. Tucker, Wreath products and proportions of periodic points,Int. Math. Res. Not. IMRN13 (2016), 3944–3969

    2016

  22. [30]

    Kantor and A

    W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group,Geom. Dedicata 36 (1990), 67–87

    1990

  23. [31]

    Kesten, Full Banach mean values on countable groups,Math

    H. Kesten, Full Banach mean values on countable groups,Math. Scand. 7 (1959), 146–156

    1959

  24. [32]

    Liebeck and A

    M.W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2,3)- gener- ation problem,Ann. Math. 144 (1996), 77–125

    1996

  25. [33]

    Liebeck and A

    M.W. Liebeck and A. Shalev, The probability of generating a finite simple group,Geom. Dedicata 56 (1995), 103–113

    1995

  26. [34]

    Odoni, On the Prime Divisors of the Sequencewn+1 = 1 +w1

    R.W.K. Odoni, On the Prime Divisors of the Sequencewn+1 = 1 +w1 . . . wn, J. London Math. Soc. 32(2) (1980), 1–11

    1980

  27. [35]

    Radi, A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension, arXiv preprint: 2501.00515

    S. Radi, A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension, arXiv preprint: 2501.00515

    arXiv

  28. [36]

    Radi, Groups of finite type: classification and structural properties, in preparation

    S. Radi, Groups of finite type: classification and structural properties, in preparation

  29. [37]

    Šunić, Hausdorff dimension in a family of self-similar groups,Geom

    Z. Šunić, Hausdorff dimension in a family of self-similar groups,Geom. Dedicata124 (2007), 213–236

    2007

  30. [38]

    Uria-Albizuri, On the concept of fractality for groups of automorphisms of a regular rooted tree, Reports@SCM, 2 (2016), 33–44

    J. Uria-Albizuri, On the concept of fractality for groups of automorphisms of a regular rooted tree, Reports@SCM, 2 (2016), 33–44. Jorge Fariña-Asategui: Centre for Mathematical Sciences, Lund University, 223 62 Lund, Sweden – Department of Mathematics, University of the Basqu...

    2016

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.